Wednesday, March 20, 2013

Concept 2 Identities and Right Triangles





This image shows ratios being solves with identities.




This picture shows ratios being solved with SOH CAH TOA.


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This image shows ratios being solved with Unit Circle Ratios. These triangles fall in the quadrant 1, since we have seen in all the pictures the ratios are positive and in quadrant one the ratios are always positive.




Tuesday, March 19, 2013

Different Levels: Unit Q Concept 4


        These pictures explain the different levels of Unit Concept  4





In each level we have to remember to give an answer in degrees and ratios.

Deriving Pythagorean Identities



1. The first Pythagorean Identity comes from the unit circle by drawing out a right triangle and using all three sides of it: the hypotenuse, opposite side, and adjacent side. The hypotenuse equals one (unit circle radius always equals 1), we will label the opposite side with Y and the adjacent side with X. Now we plug this into the Pythagorean Theorem Formula as Y^2+X^2=1.  Now we just substitute the Y with Sinᶿ and the X with Cos ᶿ, this leaves us to get our Pythagorean Identities Equation. The Pythagorean Identity equation is  Sin^2ᶿ+Cos^2ᶿ=1.

This video I included shows the derivation of the  First Pythagorean Identities followed by the second and third Pythagorean Identity. I included this video because it  goes step by step using the Unit Circle Ratios and Reciprocal Identities to get the all  Pythagorean Identities.  Enjoy!!



The first identity is the base of the Pythagorean Identities because it helps us find the other two Identities.  To get to our second identity involving tanᶿ and secᶿ we need to divide the first identity with cosine.

Sin^2ᶿ+Cos^2ᶿ=1               Sin^2ᶿ  +   Cos^2ᶿ   =   1      Now we use ratio identities and
       Cos^2ᶿ                        Cos^2ᶿ       Cos^2ᶿ   Cos^2ᶿ   reciprocal identities to obtain the 
                                                                                              second identity

In the end you get: tan^2ᶿ+1=sec^2ᶿ                            Don’t forget to power up the identities.

In getting the third identity we need to divide the first identity with sin.

Sin^2ᶿ+Cos^2ᶿ=1      Sin^2ᶿ  +  Cos^2ᶿ    =  1                      We use ratio identities and reciprocal
    Sin^2ᶿ                  Sin^2     Sin^2ᶿ      Sin^2ᶿ            reciprocal identities to obtain the 3rd identity                                                                                               

      In the end you get: 1+Cot^2ᶿ=Csc^2ᶿ

Sunday, March 17, 2013

Math Analysis Reflective Blog Post


1.      1.     How have you performed on the Unit O and P tests?  What evidence do you have from your work in the unit that supports your test grade (good or bad)?  Be specific and include a minimum of three pieces of evidence.

RESPOND HERE:
               My Unit O and P test performance haven’t been my best so far this year. I received a 52% the first time I took the Unit O test which is my worst grade I have received the whole year. Though I received a low grade I was given another opportunity to retake the whole test for a better grade. I am grateful that I had the chance, but my performance once again wasn’t what I expected. I received a 73% on my retake which is a whole letter grade better than how I did last time, but I still didn’t feel that was good enough. I was actually was surprised I received this grade because I studied a lot and felt confident, but I received a score that I wasn’t really expecting. For example I re-watched all the videos again and did extra problem to get ready for the retake.  Moving on to my Unit P performance I received a 76% on my test. I understand the grade I got because I did foolish mistakes on the test. For example I used the numbers for problem number two as problem number four.  
2. You are able to learn material in a variety of ways in Math Analysis.  It generally follows this pattern:

→ Your initial source of information is generally the video lessons and SSS packets followed by a processing and reflection activity via the WSQ
→ individual supplemental research online or in the textbook before class
→ reviewing and accessing supplementary resources provided by Mrs. Kirch on the blog
→  discussion with classmates about key concepts
→ practice of math concepts through PQs
→ formatively assessing your progress through concept quizzes
→ cumulatively reviewing material through PTs
→ Final Assessment via Unit Test.

Talk through each of the steps given in the following terms:
a. How seriously do you take this step for your learning?  What evidence do you have to support your claim?  Make sure to make reference to all 8 steps.
b. How could you improve your focus and attention on this step to improve your mastery of the material?  What specific next steps would this entail?  Make sure to make reference to all 8 steps.

RESPOND HERE:
a.       In this class I know that I have to take it seriously because this course is honors. I took the choice to being in this class. If I took this decision I had to know that I needed to be prepared with the workload and method of learning I was going to be receiving. The SSS packets and the videos that go with it is something that I take very very seriously. I take it seriously because it is the base of the math class, if I don’t watch the videos I don’t know what we are learning. Doing the WSQ is helpful because it helps me understand more to the video. For example if I missed a part of the video and there was a question about it in the WSQ I would go back to the video and watch for what the WSQ question is asking. Doing this makes me understand the part that I accidently didn’t see or understood. Included in the blog there are supplemental videos or links that are provided to further help in the learning of the Unit. These are things that I occasionally look at when I am having trouble understanding the unit. For example in Unit O I forgot how SOCCAHTOA went and I looked on the supplemental videos and links to help me review it. They were very helpful because I understood and remembered how to solve them. Another thing I do to research something when I don’t understand is looking at YouTube videos that go with the Unit we are learning. Some help me clarify questions that I had or I find shortcuts on how to solve the problems. Another outer source I use is going to tutoring with my friend who is currently in calculus. She has helped me throughout the year with helping me understand simpler ways in doing my math and has motivated me to do well. Now as for class work such as the PT’s and PQ’s I do take those extremely seriously because this is my review for the test.  For example if I am stuck on the PQ’S I ask my group members for help and if we don’t understand it we got to the U with Mrs. Kirch to get assistance with solving them. As well as using these methods for help I ask my tutor for help to clarify each step in solving the problems. In the beginning of the year I would stay every Wednesday to tutoring because I had started the year of with the wrong foot in math. We would go over Units A-D to get ready for the big test. We would even stay on library late nights for about 3 hours to review for the test.  The WPP’S and blog post are something that has helped me with the test. For example recently my partner for the WPP did a problem that I had trouble understanding and thanks to her choosing that particular problem for her WPP  I understood it and got it right on the test. The quizzes are something I have trouble with. I am going to be honest here I fall under category of taking the quiz when I am not really prepared and not truly taking it when I know the concept. When doing this I am acting like a cheese bucket student and am doing myself wrong because it’s just affecting not only my quiz grade but my test grade as well. This is an area of improvement where I need to work on.
b.        Improving each step is something I need to focus on now. Especially with a tough unit ahead I need to start trying even harder to focus on my work. Listening carefully and pausing the video when her notes pop up to read is something I can improve on. This will help when answering the WSQ because I would know how to answer the question more extensively. Not only will improving my skills for watching the video help me on  my WSQ answers it will also help me understand the unit more and it would also help with solving my PQ’S and PT’S.  The supplemental videos is something I could look at in every Unit instead of only when I need help and taking time to look at them might even help me with improving my quiz and test grades. I could use it for some extra practice.  Next doing my PT’S and PQ’S without the assistance of my SSS packet is something I am going to work on for the Units ahead. Doing this will prepare me for the test because when I take the test I don’t have the SSS packet next to me I have it all in my head. I need to memorize any formula, and each steps taken to solve the problems. Communication is a key element in this classroom; I need to communicate with Mrs. Kirch and my group whenever I am struggling with something. This I am doing great with because if I need help with something I go out there to get it answered. Since there will be no quizzes, it is being replaced with more blog work. Blog work is something that helps with showing my personal work and it depicts my understanding of it. This also helps me focus on difficult problems, by focusing on difficult problems I mean that I try and do my blog work on the problems I don’t understand. For example recently the lasts WPP’S I chose problems that I had trouble with in doing so I understood how to work the problems out. As for quizzes if we do take some for future units I will review my notes more and take it when I truly understand the concept. All in all I will try even harder to get to my goal of achieving an A in the class. Even though the units ahead will be tougher I will try my absolute best to get my work done in the best way.  


3. Reflect on your learning this year thus far by considering the following questions:
a.  How confident do you generally feel on the day of a Unit Test?  Give evidence and specifics to back up your answer.
b.  How well do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.
c.  How DEEPLY do you feel you have learned the math material this year as compared to your previous years in math?  Give evidence to support your claim.
d.  Do you normally feel like you understand the WHY behind the math and not just the WHAT/HOW?  Meaning, do you understand why things work, how they are connected to each other, etc, and not just the procedures?  Explain your answer in detail and cite specific evidence from this year.
e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life?

RESPOND HERE:
This year math has been really great!! I was exposed to a new method of learning called the Flip Classroom. I am going to be honest, at first I wasn’t so fond about this method, but as time went by I grew to love this method of learning. This method has changed the way I have been taught from the previous years in math. I received good grades in my passed math classes; passing with A’s all three years with having learned from the traditional teacher lectures. My previous math teachers really focused on teaching the math the best way they could, I believe they did succeed because I understood the concepts they taught me. They were awesome teachers, but I feel like Mrs. Kirch takes higher standards with teaching. Though I passed with A’s when it came to test questions that were review I would have only have notes to look back, but sometimes I couldn’t figure out what I had written and I would end up using YouTube as my review and  with the flipped classroom I have videos to go over and review for the test. Another thing is that this year with the flipped classroom the teacher can help the students one on one whereas for my previous years I would have to go to tutoring after or before school because class time would run out for questions. This year since I already know what lesson we are focusing on I come into the classroom prepared to ask questions. Even when I am absent I know what lesson we are going to be working on because I have my WSQ chart to know what I will be doing for the Unit. Next, understanding math analysis has been difficult, but I try my hardest to apprehend it. In the beginning it was tough because I had to get used to the flipped classroom. I started off the year with a low grade and worked my way up to earn a B in the class. Coming from a CP Algebra II class I believe it is a pretty good grade to earn though I wished I could have received an A. This just pushes me to try harder for next semester and earn an A in the class.  As for understanding what I have been learning can be hard at times, but I try my best to understand it. For example knowing how the Law of Cosine and Law of Sines came from was difficult to understand and derive, but when Mrs. Kirch explained it to our group I understood where it came from. Furthermore in Unit Q there are a lot of questions of why: Why does this formula have to be like this? or Why does that go there? For example in the beginning of Unit Q we had to explain where the ratios of tangent and cotangent came from and using the trig ratios I figured it out.  I understood this very well because I practiced what the Unit Circle and what ratios went along with it. This is a great example of why the flipped classroom is helpful because if I didn't understand the trig functions I could have gone back to review them.  Moreover test taking is something I struggle with in the class. I do feel confident when taking test, but when I feel confident my test grade turns out to be bad. Throughout the year I have received grades ranging from an F to an A, but most fall in the average category of a high C to mid B.  Each unit I study to get a good grade, but I have bad test taking skills. By bad test taking skills I mean that when I see the test I freak out and struggle with focusing on my work. Though this happens I try to calm down and remember what I was taught from the videos. In the end I feel like I have grown since the beginning of the year. As said previously I started the class with a low grade (D) and moved it to a B. From this I learned that hard work does pay off because I was chosen to be Student of the Six weeks which is something I am very grateful about.



Sunday, March 3, 2013

Math Mistakes Link: http://mathmistakes.org/?p=933


What went wrong?

The mistake that occurred was that the person trying to solve the problem did not cross multiply correctly. They did seem to cross multiply, but only one side which was the 4.21 × sin(30.4) and then they equaled it to the answer they got(2.13). This was where the mistake occurred because they were supposed to cross multiply both sides. Then they had to divide the 1.43 to both sides which then would have led them to getting 1.48 and finally to the answer which was no possible answer.Since they started off the problem wrong it led them to having all the problem wrong.

What they should have done: